Year: 2026
Type: preprint
Abstract: The classification of block-transitive 5-designs remains an open problem. The additional parameter condition that $k$ divides $v$ is called the Camina-Gagen condition. In this paper, we investigate block-transitive simple $5$-$(v,k,2)$ designs satisfying the Camina-Gagen condition. Using the classification of finite $2$-homogeneous permutation groups, we consider the affine and almost simple cases separately. We prove that no such design admits a block-transitive automorphism group of affine type. For the almost simple case, up to isomorphism, there are exactly two possibilities: a $5$-$(12,6,2)$ design admitting PGL(2,11) as a block-transitive automorphism group and a $5$-$(24,8,2)$ design admitting PGL(2,23) as a block-transitive automorphism group.
Source: arXiv (Cornell University)
Authors: Hu, Huijiao, Huang, Zheng, Shen, Shouqiang
Language: English
Topic: Finite Group Theory Research
Subfield: Discrete Mathematics and Combinatorics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7214949353